3D Manipulator Axis Constraints not working
I found the problem. I'm not sure I have all the exact details, but in essence, the problem had to do with the scaling and dimensions within my model file. I mentioned that the behavior I saw was as though I was clicking inside the white circle. My model was imported from an STL file where the size of the largest object was about 6cm across. In order to get the sizes to look right, I made the following settings, which led to the problem:
1) In the properties panel (the "N" window), I set the Scale to 0.001 for each object in the model
2) In the Properties Editor (not the N window), on the Scene tab, I set Unit Scale to 1.0.

The solution I found was to reset the individual object scale values to 1.0, and then under Properties>Scene, I set the Unit Scale to 0.001. After making this change, the net size of things on the screen is the same, but Blender seems happier dealing with objects at 1.0 scale rather than .001 scale, and the 3D manipulator now works as expected. This is probably the more correct way to do things, and I learn best by example, so I'll remember this. I just wonder if maybe there's a bug in Blender because it seems that the effective size of the little white circle in the middle of the 3D manipulator is larger than the size on screen, like it's not following the object scale properly. Anyway, I hope my answer helps others at some point. :-)